Hierarchical and size dependent mechanical
properties of silica and silicon nanostructures
inspired by diatom algae
MASSACHUSETTS INSTITUTE
OF TECHK O
EY
by
SEP 2 ' 2010
Andre Phillipe Garcia
B.S., Civil Engineering and Environmental Engineering,
Summa Cum Laude, University of South Florida (2008)
ARCHIVES
Submitted to the Department of Civil and Environmental Engineering
in partial fulfillment of the requirements for the degree of
Master of Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2010
@
2010 Massachusetts Institute of Technology. All rights reserved.
.......
Department of Civil and Environmental Engineering
t 18, 2010
A uthor ............
Certified by ..................
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Markus J. Buehler
Associate Professor of Civil and Environmental Engineering
Thesis Supervisor
Accepted by .....................
Daniele Veeneziano
Chairman, Departmental Committee for Graduate Students
Hierarchical and size dependent mechanical properties of
silica and silicon nanostructures inspired by diatom algae
by
Andre Phillipe Garcia
Submitted to the Department of Civil and Environmental Engineering
on August 18, 2010, in partial fulfillment of the
requirements for the degree of
Master of Science in the field of Civil and Environmental Engineering
Abstract
Biology implements fundamental principles that allow for attractive mechanical properties, as observed in biomineralized structures. For example, diatom algae contain
nanoporous hierarchical silicified shells that provide mechanical defense from predators and virus penetration. These shells generally have a morphology resembling honeycombs within honeycombs, meshes, or corrugated folds, and are surprisingly tough
when compared to bulk silica, which is one of the most brittle materials known.
However, the reason for the enhanced mechanical properties has remained elusive.
Here, it is proposed that one reason for the superior mechanical properties lies in the
geometric arrangement, size, and shape of the structures. By carrying out a series
of molecular dynamics simulations with the first principles based reactive force field
ReaxFF, it is shown that when concurrent mechanisms occur, such as shearing and
crack arrest, toughness is optimally enhanced. This occurs, for example, when structures encompass two nanoscale levels of hierarchy: an array of thin walled foil silica
structures, and a hierarchical arrangement of foil elements into a porous silica mesh
structure. For wavy silica, unfolding mechanisms are achieved for increasing amplitude, and allow for greater ductility. Furthermore, these deformation mechanisms
are governed by the size and shape of the structure. The ability to transform multiple mechanical properties, such as toughness, strength, and ductility, is extremely
important when looking into future applications of nanoscale materials. Altering the
mechanical properties of one of the most brittle and abundant minerals on earth,
silica, allows a new window of opportunity for humanity to create applications and
reinvent materials once thought to be impossible. The transferability of the concept
allowing for massive transformation of mechanical responses, such as brittle to ductile or weak to tough, through geometric alterations at the nanoscale, is a profound
discovery that may unleash a new paradigm in the way materials are designed.
Thesis Supervisor: Markus J. Buehler
Title: Associate Professor of Civil and Environmental Engineering
Acknowledgments
I would like to express my eternal gratitude for those who have influenced, guided,
and educated me toward my ever expanding goals. My advisor, Professor Markus J.
Buehler, is an inspiring mentor and a constant catalyst in my academic evolution,
and for this I am deeply grateful. I would like to thank Dipanjan Sen for his fundamental mentorship and friendship. I appreciate all of my lab mates for their great
companionship, and thank all those who have provided fruitful discussion and help:
Sinan Keten, Raffaella Paparcone. Zhiping Xu, Zhao Qin, Andrea Nova, and Steve
Cranford. Also, I am grateful for my family, a cornoerstone that supports me in all
endeavors, challenges, and goals.
This research was funded by the National Science Foundation through a Graduate
Research Fellowship, the Army Research Office (grant number W9-11NNF-06-1029),
the Gates Millennium Scholars Program, and a graduate research fellowship awarded
by the Department of Civil and Environmental Engineering at the Massachusetts
Institute of Technology. Their support is greatly appreciated.
Dedicado a mi padre, Felipe Garia.
6
Contents
1
2
Background
...................................
1.1
The diatom ......
1.2
Mechanical properties of nanostructures similar to diatoms . . . . . . .
1.3
Outline............................................
Methodology
2.1
Atom istic m odeling
2.1.1
2.2
. . . . . .. . . . . . . . . . . . . . . . . . . . . . . .
M echanical analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
Reactive Force Field: ReaxFF . . . . . . . . . . . . . . . . . . . . . . . .
3 Size dependence of the mechanical properties of nanoporous silicon
structures
4
. . . . . . . . . . . . . . . . . . .. . . . .. .. .. . . .
3.1
M odel geom etry
3.2
Simulation approach.......
. . . . . . . . . .. . . . . . .. .. . . .
3.3
Results and discussion......
. . . . . . . . . .. .... .... ....
3.4
C onclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
Size and hierarchy dependence of the mechanical properties of nanoporous
45
silica structures
.........
4.1
Model geometry
4.2
Simulation approach .......
4.3
Results and discussion . . . . ..
4.3.1
Surface reconstruction
. . . . . . . . . . . . . . . . . . . . . . .
45
. . . . . . . . . . . . . . . . . . . . . . .
46
. . . . . . . . . . . . ...
48
.. . . . . . . . . . . . . . . . . . . . . . .
57
. .
....
4.3.2
Preliminary investigation on the impact of the mechanical re. ..
59
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
sponse from termination of silica......... . . . . . .
4.4
5
6
65
Superductile, wavy silica nanostructures
5.1
M odel geom etry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
5.2
Simulation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
5.3
Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
5.4
Conclusion......... . . .
74
. . . . . . . . . . . . . . . . . . . . . ...
77
Conclusion
6.1
Summary of findings....................................
6.2
Discussion and future research........ ..
77
.....
........
.. .
78
List of journal publications
1. A. P. Garcia, M. J. Buehler, "Superductile, wavy silica nanostructures inspired
by diatom algae", in submission.
2. D. Sen, A. P. Garcia, M. J. Buehler, "Mechanics of nano-honeycomb silica
structures: A size-dependent brittle-to-ductile transition", in review.
3. A. P. Garcia, D. Sen, M. J. Buehler, "Hierarchical silica nanostructures inspired by diatom algae yield superior deformability, toughness and strength",
accepted for publication in: Metallurgical and Materials Transactions A.
4. A. P. Garcia, M. J. Buehler, "Bioinspired nanoporous silicon provides great
toughness at great deformability", ComputationalMaterials Science, 48(2):303309, 2010.
10
List of Figures
1-1
Variations in diatom species . . . . . . . . . . . . . . . . . . . . . . . . .
18
1-2
From fortress to diatom.
. . . .. . .. . . . . . . . . . . . . . . . . . . .
19
1-3
Diatom frustule topologies. . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1-4
Hierarchical structure of diatoms . . . . . . . . . . . . . . . . . . . . . .
21
1-5
Size scale effects on mechanical properties.
2-1
Molecular dynamics description . . . . . . . . . . . . . . . . ... . . . . .
26
2-2
Development of computing power . . . . . . . . . . . . . . . . . . . . . .
27
2-3
Flow chart of molecular dynamics . . . . . . . . . . . . . . . . . . . . . .
28
2-4
Time and length scales associated with different computational tools
.. . . . . . . . . .
used in modeling materials . . . . . . . . . . . . .
. . . .
3-1
Setup schematic and geometry variations
3-2
Stress strain graph . . . . . . . . . . . . . . . . . .
3-3
Von Mises stress field in the elastic regime, at 7% strain . . . . . . . .
3-4
Von Mises stress field at the maximum stress . . . . . . . . . . . . . . .
3-5
Von Mises stress field during failure . . . . . . . . . . . . . . . . . . . . .
3-6
Histogram plots of the Von Mises stress . . . . . . . . . . . . . . . . . .
3-7
Effect of wall width on the plastic regime, toughness, maximum stress,
and elastic m odulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-1
Geometry and setup of the hierarchical bioinspired silica structure . .
4-2
Stress-strain graph of silica foil and mesh structure
4-3
Von Mises stress field during failure, for foils . . . . . . . . . . . . . . .
. . . . . . . . . . .
22
. . . . . . . . . . . . .
4-4
Von Mises stress field during failure, for meshes
4-5
Shear stress oy taken at maximum stress for the system with wall
53
w idth w = 52 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
4-6
Von Mises stress field during failure for different mesh wall widths . .
55
4-7
Toughness map, CTOD, and competing mechanisms of shear and crack
formation.......... . . . . . . . . . . . . . . . . . . . . .
4-8
. . .. .
56
silica mesh, silicon mesh, and silica foil structures showing the effect
of wall width on the plastic regime, toughness, maximum stress, and
ductility .. . . . . . . . . . . . . . . . . . . . . . . .
4-9
. . . . . . . . . . .
58
Stress strain graph comparison of terminated and non terminated silica
fo ils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4-10 Stress strain graph comparison of terminated and non terminated silica
m eshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-1
62
Wavy topology of Ellerbeckia arenariaand corresponding silica structure used in our simulations . . . . . . . . . . . . . . . . . . . . . . . . .
67
5-2
Stress-strain graph of silica wave structures . . . . . . . . . . . . . . . .
69
5-3
Toughness, Ductility, and Maximum stress, and modulus maps of silica
wave structures........... . . . . . . . . . . . . . . . .
. . .
. . .
5-4
Von Mises stress field for structures at maximum stress and at failure
5-5
Performance map..... . . . . . . . .
5-6
Schematic of stress strain response for tensile deformation of different
morphologies of silica waves..............
. . .
. . . . . . . . . . . . . . .
. . . . . . . ....
70
71
72
73
List of Tables
1.1
Comparison between structural materials and diatom regions . . . . .
17
4.1
Comparison between H terminated and Si surface foils . . . . . . . . .
60
4.2
Comparison between H terminated and Si surface meshes
. . . . . . .
60
14
Chapter 1
Background
Since the dawn of civilization, nature been a source of countless inspiration. The
impetus for discovery in many notable minds of the past, like Newton, da Vinci,
and Darwin, has centered around nature and its fascinating characteristics. With
the unfolding of time, nature has left its indelible impact in the design of man made
structures. Presently, the field of biomimetics has come into the age of the nanoscale,
offering an impressive array of inventions that redefine conventional thought. In this
thesis, a systematic analysis of the mechanical properties of diatom inspired nanoscale
mineralized structures is undertaken. As the future foundation of synthetic materials
will rest on the shoulders of the nanoworld, this thesis will hopefully serve as a
guiding light and motivation for the use of nanoscale minerals in conventional design
paradigms and applications. The awe, magnitude, and beauty of the nanoworld is
aptly embodied by the following quote:
"To see a world in a grain of sand, And a heaven in a wild flower,
Hold infinity in the palm of your hand, And eternity in an hour
...
-
William Blake [37]
1.1
The diatom
Although invisible to the naked eye, yet unconsciously implemented in structures
throughout human history, diatoms have served as a silent backbone to human
civilization and play a role in cement, carbon sequestration, water filters, and oil
[58, 31, 17, 28, 4, 56, 71]. The impressive arrays of diatom types and applications
are shown in Figures 1-1, 1-2, 1-3, and 1-4 . But why have they served such an
important role? The answer may lie at the nanoscale. Diatoms are micrometer sized
algae with silicified, porous shells. Generally, these pores and surrounding walls have
nanometer to micrometer dimensions and serve to protect the organism and sustain
multiple biological functions. Examples of mechanical protection include preventing
virus penetration, crushing from some predator's mandibles, or digestion [73, 38, 30].
Rather few experimental data has been collected on the mechanical properties
of diatom shells, called frustules.
Hamm et al. [39] used a glass needle to load
and break diatom frustules in order to probe their mechanical response at failure.
Their findings for the F. kerguelensis species revealed an elastic modulus of 22.4
GPa, and a maximum stress along the costae of 0.6 GPa (in tension) and 0.7 GPa (in
compression). A comparison of the mechanical properties between different structural
materials and diatom regions is found in Table 1.1. Other studies [60] have used AFM
nanoindentation to study the nanoscale material properties of the porous frustule
layers of diatoms, identifying pore sizes on the order of several tens of nanometers
at the smallest levels in the hierarchy, with ultra-thin silica walls on the order of
several nanometers. For Coscinodiscus sp. It was found that the porosity increased
from the outer to the inner membrane. The pore size ranged from 45 ± 9, 192 ±
35, 1150 ± 130 nm for the cribellum, cribrum, and areola layers, respectively. Of
the three layers, the cribellum had the lowest hardness and elastic modulus: 0.076
± 0.034 GPa, and 3.40 ± 1.35 GPa, whereas the areola had the highest: 0.53 ± 0.13
GPa, and 15.61 ± 5.13 GPa, respectively [60, 59]. They observed that the variation
of mechanical properties between the frustule layers could be influenced by the pore
size, pore distance, porosity, and under different biomineralization processes.
The mechanical properties of diatoms and links to associated structural features
has already been discussed in the literature. For instance, the raphid diatom has a
raphe, which is a split in the frustule along its length with blunted ends, which helps
to reduce stress concentrations [38, 39]. Further, many diatoms share a hexagonal
pattern to the frustules, which feature a high moment of inertia due to the large
distance between the hexagon ledge and the hexagon centroid. A high moment of
inertia increases the capacity of the structure to withstand deformation [39]. The
material that composes the frustules is also important to the diatoms flexibility and
toughness. For example, during the initial stages of frustules growth, silaffins and
polyamines proteins co-precipitate with silicic acid found in the aqueous environment,
thus forming a composite organic material [43, 52, 84, 77, 70]. The cross sectional
shape of the outermost layer, the cribellum, are shallow domes with the tip pointing
outward and the cribrum forms another dome pointing inward, similar to egg shells
or hillocks. Connecting the base of the cribrum and areola is a wall type structure
reminiscent of an I-beam. The lower base of the wall forms the surface of the areola.
This I-beam shape is very common in macroscale engineering applications such as
in steel I-beams used in construction of buildings, and offers an extremely efficient
design due to its high moment of inertia, thus allowing for increased bending stiffness
and greater shear resistance. From a structural point of view, domes are also used
in engineering design due to their capacity to distribute compressive loads along the
wall. Overall, the review of earlier structural and mechanical analyses reveals that
diatoms are fascinating nanoscale structures that incorporate multiple engineering
design concepts, perhaps resulting in an optimization of their mechanical stability.
Strength (GPa)
Young's Modulus (GPa)
cement (compression)
steel
Aluminum
Copper
Silicon
Silica
cribellum
20- 55
10 -35
0.2- 1
0.1 -0.2
0.2 -0.4
0.2
0.04- 0.06
0.042 -0.11
200
70
110
180
80 -90
2.05 -4.75
cribrum
0.02 - 0.24
0.86 -2.54
areola layers
0.4 -0.66
10.48 - 20.74
Materials
diatom
and
regions
Table 1.1: Comparison between structural materials and diatom regions (Data summarized
from Ref. [60, 6]).
tnu_
e
n -11--1111
- . --=
..................................
.......
-
...........
...
. .. .......
.....
................I
.........
........
................
Figure 1-1: . These marine diatoms represent the wide array of diatom species, which
reach up to approximately 100,000. Common diatom morphologies are circular, triangular,
quadrilateral, and elliptical. Diatoms generally range from 2 - 2,000 pm in overall size [41].
Reprinted from [89].
1.2
Mechanical properties of nanostructures similar to diatoms
Other studies of the mechanical properties of brittle materials under extreme geometric confinement, in particular nanowires, revealed that by decreasing the crosssectional diameter, the material becomes stronger and is capable of undergoing significantly larger deformation before breaking [69, 49, 20]. The size scale effect on
the Young's modulus of silica wires is shown in Figure 1-5. Experimental studies
on silica nanowires of widths ranging from 230-800 nm revealed that fracture stress
was influenced by specimen size, however, the modulus was not affected by size [66].
. . ..........
Fortress: 10 m
1m
10 Cm
Figure 1-2: Merger of structure and material in engineering design [14]. The coquina
stone found in this fortress, located in St. Augustine, FL, is extremely resilient in absorbing
impacts such that cannon balls sink into, rather than shatter or puncture, attesting to its
overall ductile response even though its constituents were made of brittle materials.
This study also determined that the modulus stayed constant at around 68 GPa,
whereas the fracture stress varied from 8.77 to 6.35 GPa for the smallest and largest
SiO 2 wire widths, respectively. Ni et al. [67] experimentally determined that modulus was independent of diameter for amorphous silica nanowires ranging from 50 to
100 nm in diameter. On the other hand, Silva et al. [80] found increasing stiffness
for lower diameters by performing molecular dynamics (MD) simulations on amorphous silica nanowires. The fracture toughness of bulk vitreous silica was determined
to be approximately 0.8 MPa mi/
2
[61].
The failure mechanisms occurring within
the process zone at the crack tip in amorphous silica glass was studied by Bonamy
et al. [8].
By implementing AFM experiments and MD simulations, the authors
found that nucleation and growth of cavities are the dominant mechanism for failure
within the process zone. Integrated silicon circuits have been manufactured with a
wavy structural layout and 1.7 im thickness, and can be elongated up to 10% [48].
Other studies revealed highly ductile amorphous silica nanowires, by using a taperdrawing process. Ranging from 20 nm in diameter and with highly smooth surfaces,
the nanowires achieve extreme flexibility such that rope-like twists and spiral coils
are realized [55]. However, even though these synthetic nanostructures offer attractive mechanical properties, they are generally unfeasible for mass production due to
..
..........
(a)
...
. . .......
.......
..............
.................
(D3)
/\
Foramen
Figure 1-3: Hierarchical structure of diatoms, showing their porous silica structure. Panel
(a): Schematic of the centric diatom frustule, showing the three porous silica layers lying
along a hexagonal grid. Panel (b) shows AFM and SEM images of various diatom species,
revealing their porous structural makeup. Scale bar: 10 pm. Images reprinted from [58].
complex and expensive manufacturing. For this reason, a second look at harnessing
biomineralization is necessary. The first steps have already been taken by genetic
sequencing of certain diatoms, allowing a better understanding at what proteins are
responsible for the intricate nanoscale shapes seen on the frustules [5]. As the ability
to create these magnificent structures increases, so does the necessity to understand
the fundamental mechanical properties they endow. In light of this, we focus on a
range of mineralized nanostructures found in diatoms and use atomistic simulations
to probe the mechanical response and failure mechanisms. We hypothesize that the
nanoporous geometry of the frustules is the key to providing enhanced toughness, ductility, and strength even though the constituting material itself is inherently brittle.
Thereby, the formation of a nanostructured geometry that includes thin, geometrically confined structures of brittle elements may play a crucial role in understanding
(a)
Figure 1-4: Hierarchical structure of diatoms, showing their porous silica structure. The
SEM images in (a) show various diatom species, displaying the intricate hierarchical porous
silica wall structure of diatoms. Panel (b) shows a schematic of the hierarchical structure.
Panel (c) shows nanoscale voids observed in diatom algae, which are occluded by delicate a
silicate membrane, called a hymen (pl. hymens), which is perforated by round pores (scale
bar 200 nm). Images in (a) reprinted from [52]. Image in (b) reprinted reference [58]. Image
in (c) reprinted from [62].
the materials behavior.
1.3
Outline
This thesis focuses on the mechanical properties of nanoscale structures found in diatoms. Within this context, size and shape effects are studied for silica and silicon
systems. Indeed, diatoms are renowned for their mastery of living in harsh, mechanically extreme environments such as waterfalls and in strong sea currents. In Chapter
2, atomistic simulation methods are outlined and particular focus on the reactive
force field, ReaxFF is presented. Chapter 3 focuses on the size dependence of the mechanical properties of nano silicon mesh structures, similar to those found in diatoms.
Chapter 4 reveals the effect on the mechanical response and deformation mechanisms
from adding two levels of hierarchy implemented in nanoscale silica, foils and meshes
made up of interlocking foils, as found in diatoms. This chapter also focuses on the
effect of hydration of nano silica structures and its mechanical response. In Chapter
5, geometric effects of amplitude and width on the mechanical response and defor-
mation mechanisms are obtained for folded nano silica structures, similar to those
found in the diatom Ellerbeckia arenaria. Finally, Chapter 6 provides an overview of
the fundamental importance and impact of biomineralized structures and its future
applications while emphasizing prospective research.
140
.--
120cc;
-5
100
-
-
80
60
0
40
20
0
10
100
1000
Diameter D/nm
Figure 1-5:
The size scale effect is a common phenomenon that influences mechanical
properties at the nanoscale. The graph shows the Young's modulus of silica wires versus
diameter, for different techniques: vertical experiments (m), in-plane experiment (.), resonant frequency experiments (o) [88, 21], MD simulations of nanowires and thin films (A),
and MD simulation of bulk silica (A). Figure reprinted from [80].
Chapter 2
Methodology
In this Chapter, an overview of atomistic simulations, and the reactive force field,
ReaxFF is provided.
A historical context illuminates the evolution of the atomic
theory, with an overview of the models and simulation techniques. A description of
the theoretical framework, applications, and limitations are given.
2.1
Atomistic modeling
The inception of the atomic theory dates back to approximately 400 B.C., with the
Greek scholars Democritus and Leukippus, stating that the universe is made up of
empty space and indivisible particles, or atoms [82]. They captured a compelling
description, mentioning [42]: "Atoms, divergent in form, propel themselves through
their separation from the infinite, into the great vacuum by means of their mutual
resistance and a tremulous, swinging motion. Here gathered, they form one vortex
where, by dashing together and revolving round in all sorts of ways, the like are
separated off with the like."
Indeed this represents a striking resemblance to the actual behavior of atoms in regards to their motion, attraction, and repulsion. With this description as a harbinger
to the more complex atomistic models of the twentieth century, it was not until 1957
that the first molecular dynamics (MD) simulation was published [3].
The funda-
mental principles of molecular dynamics are illustrated in Figure 2-1. Each atom is
simplified to a point representation, with trajectories in conjunction with the interactions between other atoms through interatomic potentials. By incorporating the
Newtonian equation of motion, F = ma, and thermodynamic parameters, a description of the atom's position r(t), velocity v(t), and acceleration a(t) are gained.
Molecular dynamics offers a powerful elucidation of atomistic phenomenon, and
in many respects allows for pioneering advances in understanding nanoscale material
properties, deformation mechanisms, and even biological processes. In retrospect,
the ability to describe atomic interactions has evolved in part thanks to a symbiotic
relationship with computational technology. As the dawn of the computer age set
forth with the mass production of the Universal Automatic Computer (UNIVAC) in
the 1950's, MD studies would enter a new era. For example, the first MD simulation in
1957 [3], studied the phase transformation of a 32 particle system, proceeding at about
300 collisions per hour on the UNIVAC. The amazing development of computational
processing currently allows the simulation of millions of atoms with near quantum
detail [65]. An illustration of computing power advances relative to time is shown in
Figure 2-2.
Within MD, the trajectories of the atoms are generated by applying ensembles
such as Microcanonical (NVE) and Canonical (NVT) [7]. In the NVE ensemble, the
number of moles, volume, and energy are constant. The exchange between the kinetic and potential energies describes the path of the atoms. In the NVT ensemble,
the number of moles, volume, and temperature are constant.
The energy created
by straining the material is removed with a Berendsen thermostat. The Berendsen
thermostat allows the velocities of the atoms to rescale towards the desired temperature of 300 K. The velocity Verlet algorithm is used to update the velocity of
the atoms. The flowchart in Figure 2-3 lists the main steps comprising molecular
dynamics simulations.
2.1.1
Mechanical analysis
We calculate the virial stress [94] by
N
Zk mkvk~vk 3 +
S=V
+
V
f, and
where I and J = x,y,z; and N, m, v, r,
N
rk
kk,
a
(2.1)
V
V are the are the number of atoms,
mass of atom, velocity, position, force, and total system volume respectively.
The engineering strain is defined as:
E=
(2.2)
AL
LY
where LY is the initial length of the specimen, and ALy is the change in the length
along the deformation direction (which is the y-coordinate). The stress-strain curve
is then used to determine the elastic modulus, E, where o = U22 (tensile stress in the
loading direction):
A o0(7
E = -(2.3)
AE
OE
Once the stress-strain curves are determined, their integral is taken in order to determine the toughness:
Ey
(2.4)
(E) de,
=fa
where EV, and ef are the energy per unit volume and strain at failure, respectively.
A higher toughness indicates a greater ability of the material to absorb energy due to
stresses before failure (resulting for instance in a larger fracture process zone). The
Von Mises stress o is calculated as
-
oV
o yy)2 + (
-yy-
+)2(
g _
2
)2 + 6(g 2 +
C
+ +o )
,gx
(2.5)
where o-II (I = x, y, z) are the normal components of the stress tensor, and ox, UYz
and o-2, are the shear components of the stress tensor. We apply the above expression
(eq. (2.5)) for the Von Mises stress in plotting the atomic stresses throughout the
loading event.
Point representation
(a)
a(t)
r(t)
(b)
stretching
v(t)
Energy
repulsion
r
bending
rotation
attraction
equilibrium
bond distance
Figure 2-1: Molecular dynamics description. Panel (a): Atoms are represented as points
that have a certain position, velocity, and acceleration. Panel (b): Each atom has an energy
landscape and different bond terms, such as stretching, bending, and rotation between that
atoms. Figure reprinted from [11].
2.2
Reactive Force Field: ReaxFF
Over the past decades, molecular simulation tools have taken various flavors, as shown
in Figure 2-4. Many incorporate MD schemes with various types of potentials such
as Tersoff-Brenner. At the heart of MD are potential energy functions (i.e. empirical
force field (EFF) [25, 23] which depend on the location of the atoms and their respective energy contributions. The bond interactions can be described as springs (i.e.
simple harmonic equations) that describe the compression and stretching of bonds
and the bending of bond angles. The non-bonded interactions are described by van
der Waals potential functions and Coulomb interactions. The potential functions
Computing power
"Petaflop"computers
Iol
BlueGene/L
Earth Simulator
ASCI White (CA
LINUX clusters
10' atoms
IBM Almaden Spark
"Gigaflop"
102
atoms
1965
105 atoms
1975
1985
atoms
106 atoms
1995
2005
2010
)
Year
Figure 2-2: Computing power development and associated quantitative capacity to simulate atoms with simple interatomic potentials and short cutoffs. For more complex potentials, such as ReaxFF, significantly smaller number of atoms can be simulated. Figure
reprinted from [13].
. . .
.......
....
..
..............
.......
that are defined within EFF are fitted against experimental or quantum chemical
data (i.e. training sets) [23].
The total system energy is separated into different
energy contributions:
Esystem = Ebond +Eover +Eunder +Eip+Eval+Epen + Etors+ Econg +EvdWaals+EcouIomb
(2.6)
Since the force field is empirically defined, it should only be applied to systems
similar to the training sets [23, 871. Reactive chemical systems are not properly defined by EFF methods since the shape of the potential function that defines the bond
length/bond energy relationship makes it difficult to accurately determine a paramFor instance REBO, the EFF method
eter value near the dissociation limit [23].
developed by Brener, allows the dynamic simulation of reactions in systems greater
than 100 atoms. Nevertheless, REBO is limited because it excludes non-bond interactions and is based on a relatively small training set [23].
Update velocities
and positions
Max
initial
e 2 4 F wAssign
i prtcl
Fetp
particle
Se
positions
velocities
t
ale
number
force
reached?
steps
Cluaeof
false
true
Stop simulation
and analyze
data
NVT: Rescale velocities to control temperature
NPT Rescale system volume to control pressure
NVE: Do nothing
Figure 2-3: Flow chart showing the general methodology of molecular dynamics.
The development of ReaxFF [24] reactive force field offers a more accurate view of
complex chemical systems (specifically for systems undergoing large strain or hyperelasticity) and therefore offers an alternative to the Tersoff-type potentials. ReaxFF
is founded on quantum mechanical (QM) principles, such as DFT, and defines the
atomic interactions with
QM
accuracy [24]. ReaxFF has been shown to more accu-
rately describe the material behaviors (e.g. the bond breaking and formation process)
of non metals (C, 0, H, N), metals (Cu, Al, Mg, Ni, Pt), semiconductors (Si), mixed
Si-O systems, silica-water interfaces [78, 24, 26, 83, 68, 10, 19, 18, 29], and biological
materials [12]. This aspect is critical to describe the properties of materials under
large deformation. The ab initio calculations are quite computationally expensive and
thus limit ReaxFF to systems containing a few thousand atoms. The computational
time required by ReaxFF is mostly determined by the complexity of the mathematical
expressions and the necessity to perform a charge equilibration (QEq) at each iteration [72, 9]. As a comparison, ReaxFF is approximately several orders of magnitude
faster than quantum mechanics based on first principles (ab initio) methods. However,
ReaxFF is 20-100 times more expensive computationally than simple empirical force
fields such as CHARMM, DREIDING, or covalent type Tersoff's potentials [9]. The
basic idea behind ReaxFF is to modulate the bond properties (i.e. bond order, bond
energy, and bond distance) and therefore properly dissociate the bonds to separated
atoms [24]. The valence angle terms are bond-order dependent, thereby making their
energy contributions disappear when bonds break. Upon bond dissociation, the bond
terms all smoothly go to zero and therefore no energy discontinuities appear during
the reaction [24]. ReaxFF also describes the non-bond interactions between atoms
by incorporating Coulomb and van der Waals potentials. Indeed, ReaxFF serves as
a link between QM and empirical force fields [12, 9].
........
...
...
..
. ...........
Time scale
min
Nanoindentation
Micropipette
MEMS testing
S
OpticaUmagnetic tieezers
P-S
Atomic Force Microscopy
ns
PS
Tomography
F
F-.'
A
I
x-ray
diffra ction
nm
NMR
DNA
polypeptides
I
nanopartides
I
(nanowires,
carbon
nanotubes
secondary protein
structures (e.g. betai-sheets,
alpha-helices)
Im
Ipm
Pr
Length scale
I
cels
tissues
organs
organisms
Figure 2-4: Time and length scales associated with different computational tools used in
modeling materials. One important concept lies in the coupling of different computational
and experimental tools which enable scaling up in terms of system size and time. For
example, certain reactive molecular dynamics, such as ReaxFF, utilize principles derived
from quantum mechanics and experiments. Continuum models reach the largest time and
length scales, and can incorporate atomistic results. Experimental techniques, such as those
in red font, enable the validation and provide data for different computational methods. In
this thesis, the method used is ReaxFF, a type of reactive molecular dynamics. Figure
reprinted from [16].
Chapter 3
Size dependence of the mechanical
properties of nanoporous silicon
structures
Indeed, diatoms are fascinating nanoscale structures that incorporate multiple engineering design concepts, perhaps resulting in an adaptation to improve their mechanical stability. We hypothesize that the nanoporous geometry of the frustules is
the key to providing enhanced toughness even though the constituting material itself
is inherently brittle. Thereby, the formation of a nanostructured geometry that includes thin, geometrically confined struts of brittle elements may play a crucial role
in understanding the materials behavior.
Here we utilize the porous structure found in diatoms to develop a bioinspired
nanoporous material implemented in silicon, as shown in Figure 3-1. The material
consists of a porous structure with connected truss elements, where the width w of
the nanotrusses is controlled in our analysis.We note that the goal of this study is not
to develop a model that very closely reflects the specific structure of diatoms. Rather,
our approach is guided by our desire to develop a general model system in which we
can test the effect of the width of the nanotrusses on the bulk material behavior.
The choice of silicon is motivated by the fact that a good interatomic potential,
the first principles based reactive force field ReaxFF, is available for this material,
which has been validated against experimental results for mechanical properties, in
particular large deformation and fracture of silicon [15, 10]. By varying the size of the
constituting silicon nanostructure, we examine associated mechanical properties, as
well as fracture and toughening mechanisms, facilitated through a series of molecular
dynamics simulations in a computational materiomics framework [34].
3.1
Model geometry
We consider silicon crystals with uniform and ordered rectangular voids. All structures have an equivalent void length and height of 76 A by 33 A, respectively. The
only parameter varied here is the wall width w, which ranges from 5 A to 76 A.
Figure 3-1 shows the geometries considered here, representing different systems with
variations of the wall width, w. The number of atoms varies between 672 to 15,232
from the smallest to the largest width systems. The simulation cell has dimensions
of 217.20 A x 152.039 A x 10.86
3.2
A in the
x-, y- and z-direction.
Simulation approach
The structure is relaxed to its minimum potential energy and then loaded under
tension along the [1 0 0] direction as shown in Figure 3-1(a), at a strain rate of
1x10 10 s-1.
The system has periodic boundary conditions in all three directions,
with a constant temperature of T=300 K controlled by the Nosd-Hoover thermostat.
Deformation is applied by uniaxially increasing the size of the periodic simulation
cell, while keeping all other dimensions of the simulation cell constant. Since the
horizontal ligaments are interconnected with the vertical ones and because the poison
effect along the non-loading directions are not considered, the stress state is globally
multiaxial. We use a time step of 1 femtoseconds. The initial, unstrained silicon
structure is shown in Figure 3-1(b).
Aside from the variations in the geometry,
all simulations are carried out under identical conditions, enabling us to perform
a systematic comparison.
(a)
loading direction
zvoid
silicon
z [010]
x [100]
W
(b)
w =11 A
w= 22 A
w=43A
w= 76 A
Figure 3-1: Geometry of the bioinspired silicon structure, and setup used in our simulations. Panel (a): Three-dimensional schematic of the silicon structure, with periodic boundaries along the x, y, and z directions. The arrows indicate tensile load applied uniformly
along the structure. Panel (b): Initial geometry of structures considered here, illustrating
the wall width (w, definition indicated in one of the structures) variation in the geometry.
3.3
Results and discussion
We now present a systematic analysis of the effect of changes of the wall width w on
the resulting mechanical behavior (see Figure 3-1 for geometry). Figure 3-2 shows the
stress-strain graphs for all sizes considered here, where the wall widths range from w
= 5
A to
76
A.
For smaller wall widths, we find that a greater plastic regime, lower
maximum stress, and a lower modulus. Conversely, for larger wall widths we find a
reduced plastic regime, larger maximum stress, and a higher modulus. The behavior
of the material at larger wall widths more closely resembles that of bulk silicon, which
is inherently brittle. However, due to lowering the wall width of the structure it is
possible to dramatically change the behavior of the material towards a more ductile
fashion, and facilitate the occurrence of very large deformation up to 80% without
breaking. We ascribe the decrease in the maximum stress and the modulus with
lower wall widths from free surface effects and a change of the stress distribution in
the sample. Furthermore, the prominence of the plastic regime found for decreasing
wall widths is not solely a phenomena caused by surface effects, but one that is
augmented by the nanoscale porous architecture, which allows for a conformational
change from a rectangular to hexagonal shape due to the change in the stress state,
becoming more uniform as the wall width decreases.
We proceed with an analysis of the stress fields and the structural changes of the
nanostructure during deformation of the material. Figure 3-3 shows the Von Mises
stress field in the elastic regime, at 7% strain, for different wall widths. It can be
recognized that for smaller wall widths, the stress is more homogeneously distributed
throughout the system, and that a greater fraction of the atoms feature a higher stress
level. In larger systems, the formation of a stress concentration at the corners can
be recognized, suggesting a possible point for the nucleation of cracking or shearing.
Figure 3-4 shows the Von Mises stress field at the maximum stress, for different wall
widths. Here the strain value at which the snapshot was taken changes for the different cases considered (see stress-strain graphs shown in Figure 3-2), and the relevant
strain level is indicated in the plot. For widths smaller than 11 A, the structure at
........
........
...
. ......
w large
A.0
*5
e16A
*27A
38A
*54A
76A
e5
0.
(5
C<)
CD)
11 A
01
*22A
*33A
*43A
*65A
wsmall
I
1
~I1
1 -I
0.2
I
I
I
I1
I
IT I
I
I
0.4
0.6
0.8
Engineering Strain
Figure 3-2: Stress-strain graph for all sizes. Wall widths range from w = 5 A to 76 A.
For smaller wall widths, there is a greater plastic regime, lower maximum stress, and lower
modulus. Thus, due to the lowering of the wall width of the structure (w), the system
behaves in a more ductile fashion and sustains very large deformation up to 80%.
the maximum stress becomes hexagonal at the maximum stress, representing a significant structural transformation compared with the initial rectangular structure. It
is evident that the stress is distributed homogeneously throughout the structure. For
larger wall widths, high stresses are concentrated around the corners and in diagonally oriented regions in the sample. Moreover, the initial, rectangular shape of the
structure is maintained up to the maximum stress, suggesting a limited capacity of
the system to undergo plastic deformation. Indeed, further analysis at larger strains
(during the failure process) confirm this conjecture. Figure 3-5 depicts the Von Mises
stress field during failure for different wall widths, where again the strain value at
which the snapshot was taken is indicated in the plot. For the systems with wall
widths of 11 A and 22 A, necking and formation of beaded molecular structure is
observed, quite similar to what has been seen in earlier studies of metal nanowires.
At widths of 43A and larger, we observe that nanocracks initiate from the corners
and then arrest, leading to further necking. Shear deformation in these thinned parts
of the structure contributes to the final failure of the samples. For widths of 76 A
and similarly large systems we observe the formation of voids within the sample along
the plane of shear. The shearing direction is orientated at roughly 450 degrees with
respect to the tensile direction. Failure mechanisms in silicon nanowires examined in
earlier molecular dynamics and experimental studies show similar results as observed
in our simulations, in that the fracture process zone undergoes amorphization [40, 44].
In our simulations, the region surrounding the failed region tends to be amorphous.
We estimate that the amount of amorphization (determined from the edge where
failure takes place) extends at most 10-15
A
into the sample. Crack arrest is due
to the nanoscale geometry, and can also be explained by simultaneous cracking. For
example, for the 43 A wall width, both cracks initiate on the same ligament but on opposite edges. One crack travels through the vertical ligament (which is the dominant
crack), while the other crack travels into the adjacent horizontal ligament at roughly
450 degrees to the loading plane. The crack that travels perpendicularly through the
vertical ligament has less distance to traverse, and thus passes through the ligament
before the other crack can. Therefore, the 450 degree crack arrests since the system
-- P
MOWN"
ft
is no longer intact. For slower strain rates of 1x109 s-1, the crack arrest mechanism
still occurs, but wedge which is formed from the nondominant crack is smaller than
those from 1x10 10 s-1.
w=22A
w=11A
Von Mises stress [GPa]
w=76A
w=43A
10
8
6
4
2
0
Figure 3-3: Von Mises stress field in the elastic regime, at 7% strain, for different wall
widths. For smaller wall widths, the stress is more homogeneously distributed throughout
the system, and a greater fraction of atoms have higher stress. In larger systems, a stress
concentration at the corners can be recognized.
The analysis reviewed above suggests that the change in wall width induces a
change in the stress distribution during deformation.
This is confirmed with an
analysis of histogram plots of the Von Mises stress, as shown in Figure 3-6, where
results are shown for w
maximum stress. For w
11
A and
76
A.
Figure 3-6(a) depicts the results at the
11 A the stress distribution shows a peak at a larger
stress than at w = 76 A. Figure 3-6(b) depicts the results during failure. It is found
that the stress distribution in the w = 76
A case
is much different from the one in
the w =11 A case. For w = 11 A, the stress distribution shows a peak at a much
larger stress (6.5 GPa) compared with a peak at w = 76 A (2.8 GPa) along with
a broader tail. The broader tail in the large system indicates a more heterogeneous
distribution of the stress in the sample. Finally, in Figure 3-7 we summarize the
effect of wall width variations on various mechanical parameters. Figure 3-7(a) plots
the variation of the plastic regime, Figure 3-7(b) the toughness, Figure 3-7(c) the
maximum stress, and Figure 3-7(d) the elastic modulus. Both the maximum stress
and modulus are found to increase with the wall width. A general trend of larger
plastic regimes for decreasing wall widths is observed as well, reaching plastic regimes
approaching 40% for the smallest systems considered with a wall width of a fraction of
a nanometer. For wall widths larger than 27 A, the toughness plateaus at around at
9x108 J/M 3 . Between 16
maximum at 16
A and
A. Below
27
A a sharp
increase in toughness is observed, with a
16 A, the toughness drops to around 7x108 J/m 3 , denoting
an inverse trend. The reason for the greater variation in toughness for w = 27
A are
fluctuations in the initial deformation mechanism from necking to crack formation,
perhaps due to the fact that this system is close to the critical dimension between the
transition between the two types of extreme behavior. The results depicted in Figure
3-7 clearly show that by controlling the wall width, a significant improvement of the
material performance can be achieved. Specifically, at a critical dimension of 18
A,
the material features the greatest possible toughness. Overall, the study shows that
the bioinspired silicon nanoporous bulk material provides great toughness at great
deformability.
3.4
Conclusion
By employing an atomistic simulation method derived from first principles quantum
mechanics, we have examined the deformation and failure behavior of bioinspired
porous silicon structures as shown in Figure 3-1. The inspiration for the structural
design is based on porous silica cell walls of diatoms, unicellular algae that boast an
w =11 A
w= 22 A
Von Mises stress [GPa]
10
w=43A
w=76A
4
8
6
2
0
Figure 3-4:
Von Mises stress field at the maximum stress, for different wall widths
(the strain value at which the snapshot was taken is indicated in the plot). For widths
smaller than ~ 11 A, the structure at the maximum stress becomes hexagonal, and the
stress is distributed homogeneously throughout the structure. For larger wall widths, high
stresses are concentrated around the corners and in diagonally shaped regions in the sample.
Moreover, the initial, rectangular shape of the structure is maintained.
-
-- --------
w=11 A
E = 75.1%
w=22A
E = 71%
Von Mises stress [GPa]
10
w=43A
w=76A
4
8
6
2
0
Figure 3-5: Von Mises stress field during failure for different wall widths (the strain
value at which the snapshot was taken is indicated in the plot). For the systems with wall
widths of 11 A and 22 A, necking and formation of beaded molecular structure is observed.
At widths of 43 A and larger, cracks initiate from the corners and then arrest, leading to
further necking. Shear deformation in these thinned parts of the structure contributes to
the final failure of the samples. For widths of 76 A and similarly large systems we observe
the formation of voids within the sample, where the region surrounding the failed region is
amorphous. The failure mechanism remains similar for the wall widths below 43 A, and is
characterized by a structural change from a rectangular to hexagonal shape. An analogy to
plastic hinges can be drawn to describe the mechanism for allowing large deformation. For
larger systems, however, the failure mode is consistently crack propagation, an effect that
is confirmed to exist for varying strain rates.
M-M
MER-P
(a) 35
+11
<30
S76
400
350<
300
250$
200+-
A, E=71%
A, E 17%
1125
L20
4-
>15
150w
*10
25
U.
0
100
0
50
L
0
0
3
6
9
12 15
Von Mises Stress [GPa]
(b) 50
45
+11
e-40
E=73%
E=25%
500
450<
4000
35
350 "
3:30
300
250
2000
150
100 )
4025
>20
615
or10
u. 5
0
50"-
0
3
6
9
12 15
Von Mises Stress [GPaj
0
Figure 3-6: Histogram plots of the Von Mises stress, (a) for w = 11 A and 76 A (at the
maximum stress), and (b) for w = 11 A and 76 A during failure. Panel (a): For w = 11 A
the stress distribution shows a peak at a larger stress than at w = 76 A. Panel (b): During
failure, the stress distribution in the w = 76 A case is much different from the one in the w
= 11 A case. For w = 11 A, the stress distribution shows a peak at a much larger stress (s
6.5 GPa) compared with a peak at w = 76 A (~ 2.8 GPa) along with a broader tail. The
broader tail in the large system indicates a more heterogeneous distribution of stresses.
..
..........................................................
(a) 50,
.. . .
(b)
ie40,
1.5
E
' 1.2
E 30,
, 0.9
*D20;
..
. .
.............
....
D0.6
10
S0.3
.0
0 1020304050607080
Width [A]
)rCO)
0-.
0
(d) 40
j
0 1020304050607080
Width[A]
30
CD,
w20
E
E
x
CU
'10
0 1020304050 607080
Width [A]
0
1020304050607080
Width[A]
Figure 3-7: Effect of wall width on (a) the plastic regime, (b) toughness, (c) maximum
stress, and (d) elastic modulus. Both the maximum stress and modulus are found to increase
quasi-linearly with the wall width. A general trend of a higher plastic regime for decreasing
wall width is observed as well. The plastic regime is estimated by measuring the length of
the linear plateau region which is associated with constant stress. For wall widths larger
than 27 A, the toughness plateaus at around at 9x10 8 J/m 3 . Between 16 A and 27 A a sharp
increase in toughness is observed, with a maximum at ~ 16 A. Below 16 A, the toughness
drops to around 7x108 J/m 3 , denoting an inverse trend. The error bars are determined by
dividing the standard deviation by the square root of simulation runs (that is, the standard
error of the mean (SEM)). All simulations were repeated four times. For graphs (c) and (d),
the error bars are smaller than the data points surrounding it so they cannot be identified
in the plot.
intriguing hierarchical design and interesting mechanical properties (Figure 1-3).
Through carrying out a systematic analysis, we have captured the variation in mechanical properties, as summarized in Figure 3-7. Through the design of a nanoporous
structure with ultra-thin trusses, the material response of silicon has been altered from
brittle to highly ductile, where the most ductile systems with wall widths below 1
nm feature a plastic regime of almost 40%, and a modulus of 2.3 GPa. The most
brittle responses are observed for the systems with the largest wall width, with a
vanishing plastic regime, albeit a larger modulus of more than 30 GPa. We find that
an important aspect in promoting the ductile response of the material is the ability to alter the initial rectangular shape into a hexagonal one, which occurs for w <
33
A. Interestingly,
a Hall-Petch and inverse Hall-Petch like effect was observed for
the toughness; with a maximum toughness of 1.2x109 J/m
16
A
(see, Figure 3-7(b)).
3
corresponding to w =
In comparison, the toughness values of high-performing
engineering bulk amorphous materials range from 10 to 100
x106
J/m 3 for ceramics
and lxO106 J/m 3 for Ti alloys [45]. Plain lightweight aggregate concrete and steel
fiber reinforced concrete has a compressive toughness of 150x103 J/m
3
and 290x103
J/m 3 , respectively [46]. In light of these toughness values, the material considered
here shows a manifold increase of the toughness.
The knowledge derived from the simulations reported here heralds a paradigm
shift in the design of conventionally brittle materials by showing that its mechanical
response can by greatly altered by simple alteration of its structural geometry at the
nanoscale without the need to introduce new constituents. This merger of material
and structure is a powerful concept that could provide new ideas for a broader class
of bioinspired materials with advanced properties. Specifically, our study provides
a strategy to transform an inherently brittle material towards a very ductile one,
illustrating how a weakness is turned into a strength simply by controlling its structure. This is achieved by providing large elasticity and plasticity by adding ordered
nanopores to a brittle system, where the underlying mechanism change is due to
surface effects and a change in the stress distribution. The deformation mechanism,
incorporating structural changes in the organization of the material in the bulk sys-
tem, is unique and does not strictly resemble that of one-dimensional nanowires. The
overarching feature of diatom shell structures is their nanoscale porosity, which we use
here as a template in finding mechanical properties of similar structures. We achieve
this by applying the ordered nanoporous design concept found in diatoms shells in developing a new silicon nanomaterials where key design features are transferred. Such
a material could have many applications in the development of structural materials,
filters, optical materials, and many others. For the fabrication of the class of materials discussed here, self-assembly techniques could be used, for example based on
protein, peptide or other molecular scaffolds [1, 91]. A similar strategy is found in the
synthesis of diatoms, providing a possible path to manufacturing these materials with
great precision to reach relevant length-scales [75, 43]. The great abundance of brittle
materials in nature - such as silica or other minerals, combined with novel energy
efficient self-assembly techniques, could provide a new route for the development of
lightweight, strong and tough materials with small ecological footprint.
The studies reported here have focused solely on variations of the wall widths, and
does not address other design parameters such as the aspect ratio or the overall geometry of the nanostructure. Further analysis of other geometric variables may thus
be another promising avenue of future work and could provide additional insight in
optimizing the material. It may also be interesting to incorporate topology optimization to tailor certain mechanical properties. In the next chapter, a detailed analysis
of the mechanical properties and advantages of hierarchical structures representative
of diatom frustules is established.
Chapter 4
Size and hierarchy dependence of
the mechanical properties of
nanoporous silica structures
As diatom structures are relatively complex, reaching multiple degrees of hierarchy
and size, revealing the intricate interplay between structure and mechanical response
requires delicate and systematic analysis. The previous chapter established a framework on size effects on the mechanical response of silicon mesh structures. In order
to understand the diatom's mechanical properties more fully, we now examine hierarchical silica structures, a common feature observed in diatoms. The mechanical
properties of diatoms are quite attractive, allowing for very high toughness, a quality
that is indeed attributed to the complex hierarchical topology resembling honeycombs
within honeycombs. In this chapter we study two structures resembling those of Fig-
ure 1-4 [331.
4.1
Model geometry
We consider a structural design composed of alpha-quartz crystals. One structure is
a foil or infinitely tall thin wall, whereas the other is a mesh composed of interlocking
silica foils thus forming uniform and ordered rectangular voids. Since the simulation
box is periodic, the foil structure can be thought of as a periodic array of thin foils
with a spacing s equivalent to the mesh structure. Figure 4-1 shows the geometries
considered here. Both the mesh and foil structures are exposed to free surfaces. The
foil has free surfaces parallel to the y axis; and the mesh has free surfaces along the
x and y axis.
All mesh structures have an equivalent void dimension of 76 A by 30 A. The
only parameter varied here is the wall width w, which ranges from 5 A to 72
A for
both foil and mesh (see Figure 4-1). The number of atoms varies from 750 to 17,000
for the smallest to the largest width silica systems. The largest simulation cell has
dimensions of 151 A x 196
relaxation for 80 ps of a 31
A x 8.5 A in the x-,
A foil structure with
y- and z-direction. We performed a
free surfaces along the x and y axis.
It was found that there was negligible average stress after relaxation. This result
could be explained since surface reconstruction does not take place.
4.2
Simulation approach
All structures are equilibrated under the canonical ensemble at 300 K for a time of 10
ps and then loaded under uniaxial strain loading along the [1 2 0] direction as shown
in Figure 4-1(a), at a strain rate of 1x10 10 s-1 at 300 K. The system has periodic
boundary conditions in all three directions and the temperature is controlled by the
Berendsen thermostat [7]. Deformation is applied by uniaxially increasing the size
of the periodic simulation cell in the loading direction only, while keeping all other
dimensions of the simulation cell constant. We use a time step of 0.2 femto-seconds.
The initial, unstrained silica structure is shown in Figure 4-1(b). Aside from the
variations in the geometry, all simulations are carried out under identical conditions,
enabling us to perform a systematic comparison.
..
. .................................................................
::..:::
.....
......
...
':::::
....
.............................................
...............
(a)
Mesh
z [001]
Foil
.oi
y [120]
loading direction
x [100]
silica
w
(b)
Ir=5A
I=5
w
= 15 A,
w=
31 A
w=
72 A
Figure 4-1: Geometry of the hierarchical bioinspired silica structure, and setup used
in our simulations. Panel (a): Three-dimensional schematic of the silica mesh structure
(shown on left), with periodic boundaries along the x, y, and z directions. On the right, the
geometry of the silica foil array structure is shown, and has periodic boundaries along the
z and y directions with free surfaces only along the x axis. The spacing s between the foils
is equal to that in the corresponding mesh structure. The crystallographic orientation is
the same for all silica structures considered. The lowest level of hierarchy represented here
is the foil, and the highest is the mesh structure. The arrows indicate tensile load applied
uniformly along the structure. Panel (b): Initial geometry of mesh structures considered
here, illustrating the wall width (w, definition indicated in one of the structures) variation
in the geometry. The inset shows a detailed view of the relaxed surface structure. Wall
widths are similarly varied for the foil as in the mesh structure.
4.3
Results and discussion
We begin our analysis by focusing on the effect of changing the wall width on the
mechanical properties of the two levels of hierarchy (see Figure 4-1(a) for the two
geometries considered, that is, foil and mesh).
As shown in Figure 4-1, the wall
widths are varied between w = 5 A to 72 A for both the foil and mesh structures.
From a structural hierarchy perspective to materials design, the first order of hierarchy
consists of a foil oriented such that the only two free surfaces are in the x direction.
The second order of hierarchy is composed of a mesh of interlocking foils with voids
and free surfaces in both x and y directions. The z axis has no free surface, and thus
the structure can be described as consisting of infinitely tall walls, or foils.
Figure 4-2(a) shows the stress-strain response of the foil structure for varying
widths w, and Figure 4-2(b) shows the same data for the mesh geometry. Both the
foil and mesh structures show an increase of deformation in the plastic regime, a
lower modulus, and lower maximum stress with decreasing wall width w. However,
the silica mesh structures have a much greater plastic regime than the silica foil. The
first important observation made here is that even though silica is considered a brittle
material, the results show that it is possible to transform it into a ductile system for
small (nanoscale) wall widths, which reach a maximum failure strain of 90% and
120% for the silica foil and the mesh structure, respectively. In a similar fashion as
the foil structures, the silica mesh also shows an increased modulus and maximum
stress for larger wall widths. The plastic regime in the mesh structure decreases
with increasing wall width, albeit showing a less severe drop than in the case of
the foil structure. Interestingly, the maximum tensile stress is reached at roughly
the same strain of 34% for the silica foil, for all w
15
A
(see Figure 4-2).Another
important difference between the foil and mesh structures lies in a gradual versus a
sharp increase of elastic modulus for increasing wall width, respectively. Furthermore,
the foil structures show a stiffening effect at widths w > 15 A. Previous studies have
also shown either stiffening or softening effects, which are affected by the orientation
of the loading (since silica is anisotropic).
For example, silica nanorods modeled
using the BKS and TTAM potentials showed a softening effect when under tension
[93, 79].
Another study conducted ab initio simulations using the PAW method
and found a pressure dependence on the bulk modulus of silica [50]. Experimental
measurements of silver nanowires loaded in tension along the [0 1 1] direction found
a stiffening behavior
[90].
Copper nanowires also show either stiffening or softening,
depending on the crystallographic orientation, as shown by earlier analyses using the
EAM potential [54].
In order to gain a deeper understanding of the corresponding mechanisms that
explain the size dependent behavior of the material, we now proceed to analyze the
stress fields during deformation. We first turn to the failure mechanisms found in the
silica foil structures displayed in Figure 4-3. We find that void formation occurs for
w > 15 A, and first nucleate near the surfaces. Subsequently, voids coalesce and grow,
ultimately leading to fracture. The trajectory of void nucleation events occurs roughly
on one plane inclined to the loading axis, and for the largest w = 72 A structure, they
bifurcate onto two planes, allowing for larger regions of void coalescence. In all silica
foil structures, beading down to thin atomic chains is observed towards the end stages
of failure before complete fracture is observed. Although no clear signs of dislocation
or shear band nucleation are observed, the initial mechanisms of cavity nucleation
are consistent with other molecular dynamics simulations and experimental studies
of silica deformation [8, 76]. Previous studies on small diameter nanowires (similar
in size to those in our study) have determined that the mechanism of yielding by
dislocation is absent, and demonstrated that the main mechanical instability is a
disorder-order transformation, resulting in the reduction in size of the nanowire neck
until single atom chains emerge [49, 63].
We proceed with analysis of the silica meshes as shown in Figure 4-4, which shows
the equivalent von Mises stress field at the maximum stress. For larger wall widths,
high stress is mostly concentrated on the surface and specifically near the edges,
thus suggesting possible locations for crack or shear nucleation. However, with lower
wall widths, the stresses become relatively homogeneous throughout the structure.
For large deformation, the void shapes gradually change from a rectangular to a
940
(a)
.
MW
-
ffl
E
4
31A
jfq
25
,....
0
40
----72 A
1 small
0
0
(b)
6 wlarge
5-
0.5
Engineering strain
e5A
10 A
*21A
*15A
1 *31 A
e 52 A
*72 A
4
3
1
41 A
62 A
2
small
1
0
0
0.5
1
Engineering strain
1.5
Figure 4-2: Stress-strain graph of silica foil (panel (a)) and mesh structure (panel (b)), for
all sizes (wall widths range from w = 5 A to 72 A). Panel (a): For smaller wall widths, there
is a greater plastic regime, lower maximum stress, and lower modulus. Thus, due to the
lowering of the wall width of the structure (w), the system behaves in a more ductile fashion
and sustains very large deformation up to 115%. The strain hardening region observed for
w = 52 A is due to a temporary crack arrest due to the stretching and shearing of a
secondary horizontal ligament. The crack continues its path once the secondary ligament
stops deforming. Panel (b): For wall widths above 15 A, there exists a plastic regime of
about 1% to 5%. The greatest deformation is obtained for the smallest wall width of 5 A.
Failure mechanisms are characterized by void formations near the center or edge of the foil
which then coalesces with other voids until the structure is no longer intact. Cracks are
not observed for the foil, whereas in some mesh structures, cracks occur. The reason for
stiffening for wall widths > 15 A is due to nonlinear elasticity within the core, along with a
Poisson effect as seen in earlier studies.
-
-
_
w=15A
w=31 A
w=52A
£ = 51%
t = 51%
t = 40%
U
w=72A
t = 40% Von Mises stress [GPa]
20
16
12
4
8
0
Figure 4-3: Von Mises stress field during failure, for different foil wall widths (the strain
value at which the snapshot was taken is indicated in the plot). The Highest stresses
are localized near the failure zones and the surface. The figure also reveals the deformation
mechanisms, showing the locations of void nucleation and coalescence. Bifurcation of voided
regions and shear like behavior is observed for w = 72 A. A beading mechanism narrowing
down to thin strings of atoms is observed for the final stage in deformation. In order
to improve image clarity, we only show the stress values associated with silicon atoms
when plotting the stress fields. Generally, the oxygen atoms have a much broader stress
distribution than silicon atoms, and therefore the stress patterns are difficult to observe if
both atom types are shown.
hexagonal one for decreasing wall widths, and can be clearly seen for w
31 A. High
levels of shear stress also manifest in a diagonal pattern as shown in Figure 4-5 for
w > 31 A , which facilitate possible regions of void nucleation and the start of failure.
Once crack failure starts, the stress is concentrated around the fracture process zone,
as shown in Figure 4-6.
The deformation mechanisms observed for varying wall
widths are dramatic in that they correlate with the peaking of toughness, and can be
summarized as follows. For w > 62 A and w < 21 A, the dominant failure mechanisms
are brittle crack propagation, and beading down to thin atomic chains respectively.
However, for 21 A
w
62 A, crack propagation and shear mechanisms occur in
a competing fashion, and allow for increased toughness. For example, for w = 52
A structure, we observe a crack arrest phenomenon and corresponding increase of
tensile stress from 2 GPa to 2.8 GPa, which is due to shear mechanisms occurring
within the junctions of the mesh elsewhere (see Figure 4-7).
The analysis discussed in the preceding paragraph explains the remarkable stressstrain response of mesh structures with thin wall widths, as shown in Figure 4-2(b).
The key to explain these is the geometric pattern that allows large deformations to be
accommodated by the mesh by changing from a rectangular pattern to a hexagonal
one at large strains (see, e.g. in Figures 4-4 and 4-6), specifically for wall widths below
31 A. The fundamental reason for these very large strains without failure is due to
the more homogeneous distribution of stresses and the geometry transformation from
rectangular to a hexagonal shape for smaller wall widths.
In Figure 4-8, we summarize the effect of wall width variations and hierarchy
level on the mechanical properties - the plastic regime, toughness, maximum stress,
and ductility. In all structures considered here, the plastic regime increases with
decreasing w. The maximum stress and modulus both increase with the wall width,
and the ductility increases for smaller wall widths.
For the largest wall width in
silica and silicon structures, the range of ductility is between 30 % to 50 %. The
greatest ductility is observed for the silica mesh with smallest wall width of 5 A,
reaching 120%. The silica foil shows a gradual increase in modulus with width, with
a maximum of 6.7 GPa for w = 72 A, whereas the mesh structures sharply increase in
w=15A
w= 31 A
Von Mises stress [GPa]
20
0
16
w=72A
121
k4 a 44
w=52A
E= 31%
E = 30 %
8
4
0
Figure 4-4: Von Mises stress field at the maximum stress, for different mesh wall widths
(the strain value at which the snapshot was taken is indicated in the plot). For widths
smaller than ~ 31 A, the structure at the maximum stress becomes hexagonal, and the
stress is distributed homogeneously throughout the structure. For larger wall widths, high
stresses are concentrated around the corners. Moreover, the initial, rectangular shape of the
structure is maintained. In order to improve image clarity, we only show the stress values
associated with silicon atoms within the silica system.
...........
....
..
....
w=52A
10
Shear Stress [GPa]
5
0
-5
-10
Figure 4-5: Shear stress osy taken at maximum stress for the system with wall width w =
52 A. High regions of shear stress form a diagonal pattern and suggest possible areas where
deformation occur. We show multiple sets of the periodic cell so that the stress pattern can
be clearly seen. In order to improve image clarity, we only show the stress values associated
with silicon atoms within the silica system.
w=15A
w=31A
= 120%
Von Mises stress [GPa]
20
16
w=52A
E
62%
w=72A
=33%
125
8
4
0
Figure 4-6: Von Mises stress field during failure for different mesh wall widths (the strain
value at which the snapshot was taken is indicated in the plot). For the systems with wall
widths of 15 A and 31 A, necking and formation of beaded molecular structure is observed.
At widths of 52 A and larger, cracks initiate form the corners. For w 31 A, we observe the
formation of voids within the sample, specifically within the regions surrounding the failure
process zone. The failure mechanism remains similar for w 52 A, and is characterized by
a structural change from a rectangular to a hexagonal shape. An analogy to deformation
in mascroscopic plastic hinges can be drawn to describe the mechanism for accommodating
large deformations. For larger systems, however, the failure mode is consistently crack
propagation, an effect that is confirmed to exist for varying strain rates. In order to improve
image clarity, we only show the stress values associated with silicon atoms within the silica
system.
1.5
C.,
p1.2
0.9
r 0.6
0.3
pure
beading
shear &
crack
pure
cracking
0.0
0
20
40
60
84
Width[A]
(b)
Shear Stress [GPa]
10
5
0
a.
0-3
-5
C) S2
-101
1
0
0
0.2
0.4
0.6
0.8
Engineering strain
Figure 4-7: Panel (a): Toughness map with corresponding failure mechanism for the
silica mesh. Panel (b): Toughening and stiffening mechanisms are caused by competing
mechanisms of shear and crack formation. The crack tip opening displacement (CTOD)
measurement reveals crack arrest and is plotted against the corresponding stress-strain data
which reveal how well correlated both mechanisms are. Panel (c) shows the locations of
shear and crack formation. For purposes of clarity, only the silicon atoms are shown.
modulus, reaching 36 GPa and 29 GPa for silicon and silica meshes, respectively. The
effect of hierarchy on toughness is quite striking because the foil does not show a sizedependent toughness peaking response, as was observed for the meshes, and because
it has a consistently lower toughness than the meshes. For example, the maximum
toughness observed in silicon and silica meshes are 1.20 x 109 J/M 3 and 1.29 x 109
J/m 3 , respectively, yet the silica foil reaches only 0.60 x 109 J/m 3 . The reason for
greater toughness in the higher hierarchy of meshes lies in competing mechanisms of
shear and crack, wherein crack arrest is achieved either through shearing of another
foil subcomponent in the mesh structure (as observed in the silica mesh, w = 52
A), or through simultaneous cracking of different regions (as observed in the silicon
mesh, w = 43
A).
These competing mechanisms are enabled through the hierarchical
assembly of the foil elements into the mesh structure, and could not be achieved in
unit foil structures alone. This result demonstrates that including higher levels of
hierarchy are beneficial in improving the mechanical properties and deformability of
silica structures.
4.3.1
Surface reconstruction
Surface reconstruction may occur and may have some effect on the mechanical behavior. For example, a previous molecular dynamics study reported a stiffening of
silicon nanowires once the surfaces were reconstructed [92].
For a 1.05 nm thick
nanowire, the modulus increased from approximately 150 GPa to 160 GPa once the
[001] surface was reconstructed.
The authors attributed this stiffening effect from
bond saturation as reconstruction takes place. A previous study used ReaxFF and
observed reconstruction of a ZnO surface after 300 ps at 700 K [74]. In another study,
it was shown that a total time-scale of 240 ps was required to obtain a reconstructed
silicon surface by an annealing process with the Tersoff potential and a total system
size of 308 atoms [57]. Although it is expected that our ReaxFF based approach can
capture surface reconstruction in principle, it is computationally expensive to capture
this due to associated time-scale and system size, and was thus not observed in our
simulations. Indeed, for small systems such as 1,000 atoms, ReaxFF could feasibly
........
....
..
(a) 50
(b) 1.5
+0SiO2 Mesh
o40
G-Si Mesh
U +SiO 2 Foil
E 30
.5
2 0
e0
20
(C)
1.2
0
0.9
U)
(I)
0.6
0)
C
0)
0
F-
zcc 10
0
(V)
40 60
Width[A]
0.3
0.0
80
0
(d)
00
0
$D
0f)
100
80
60
40A
E
20
0
0
20
40
Width[A]
60
80
80
40 60
Width[A]
80
120
E
CU
20 40 60
Width[A]
e
0
20
Comparison between silica mesh, silicon mesh, and silica foil structures
Figure 4-8:
showing the effect of wall width on (a) the plastic regime, (b) toughness, (c) maximum
stress, and (d) ductility. Data for the nanoporous silicon is retrieved from [34]. Both the
maximum stress and modulus are found to increase with the wall width. The ductility
generally increases for smaller wall widths. The plastic regime is estimated by measuring
the length of the linear plateau region which is associated with constant stress. In silicon,
for wall widths larger than 27 A, the toughness plateaus at around at 9x108 J/m 3 . Between
16 A and 27 A a sharp increase in toughness is observed, with a maximum at w ~ 16 A.
Below 16 A, the toughness drops to around 7x108 J/m 3 , denoting an inverse trend. In the
silica mesh the highest toughness is observed for wall widths of 41 A, reaching values ~
1.29x10 9 J/m 3 . The silica foil generally increases in toughness with the wall width, yet has
lower toughness when compared to the mesh structures. Thus, by increasing the level of
hierarchy, a higher toughness, maximum stress, and modulus can be achieved.
simulate the time scales listed above. However, for systems larger than 10,000 atoms,
ReaxFF would be very computationally expensive. The investigation of surface reconstruction and its effects on the mechanics of hierarchical silica structures could be
an interesting subject of future studies.
4.3.2
Preliminary investigation on the impact of the mechanical response from termination of silica
When exposed to a natural environment, silica surfaces generally become terminated
with hydrogen after exposure to moisture. Results have been obtained in previous
studies of silica nanorod deformation in the presence of water using semi-empirical
quantum mechanics methods [79], where the authors concluded that strained siloxane
(Si-0-Si) bonds are attacked by water which results in lower stress and lower failure
strain of the silica nanorod, compared to a dry silica nanorod.
In the following investigation we consider two cases of silica surfaces: 1) bare
silicon surfaces that are under coordinated, herein referred to as non terminated, and
2) hydrogen bonded to a surface of oxygen, herein referred to as non terminated.
Two geometries are considered, foils of w-15
A and meshes
of w=5 A. One common
process by which surface termination is achieved is through contact with water to the
silicon surface dangling bonds of silica. The water dissociates into OH and H, with the
hydroxide bonding to the silicon. In our simulations, all structures are equilibrated
under the canonical ensemble at 300 K for a time of 10 ps and then loaded under
uniaxial strain loading along the [1 2 0] direction at a strain rate of 1x1010 s-1 at
300 K. The system has periodic boundary conditions in all three directions and the
temperature is controlled by the Berendsen thermostat [7]. Deformation is applied by
uniaxially increasing the size of the periodic simulation cell in the loading direction
only, while keeping all other dimensions of the simulation cell constant. We use a
time step of 0.2 femto-seconds. The initial, unstrained silica structure is shown in
Figures 4-9 and 4-10.
As the terminated structures are stretched, the surfaces form various types of
polymorphs with geminal and interacting silanols as the most common. The non
terminated foils have a greater maximum stress than that of the terminated foil, 1.38
GPa versus 1.21 GPa, respectively, as shown in Table 4.1 and Figure 4-9. Ductility
is reduced from 67.2% to 66.2% as the foil is terminated. The results of lowering
ductility and types of observed polymorphs are consistent with other studies [79, 64].
As the meshes are stretched, termination allows a ductility of 250% versus 122%
for non terminated, as shown in Table 4.2 and Figure 4-10. Since the terminated
meshes are fully coordinated, the struts are thinned down to atom chains with a
one silicon atom thickness that are symmetrical, whereas the non termianted meshes
form amorphous like struts that fail earlier. Also, the hydrogens atoms do not seem
to break any Si-O-Si bonds, thus offering another explanation for the hight ductility.
max stress [GPa]
strain at max stress [%]
failure strain [%]
H Terminated
1.21±0.01
35±0.5
66.2±3.6
Si Surface
1.38±0.037
48±5
67.5±5
Table 4.1: Comparison between H terminated and Si surface foils of ductility, maximum
stress, and strain at maximum stress).
max stress [GPa]
strain at max stress
failure strain [%]
[%]
H Terminated
0.84±0.008
244±4.5
Si Surface
0.69±0.023
105±5
250±4.1
122±7
Table 4.2: Comparison between H terminated and Si surface meshes of ductility, maximum
stress, and strain at maximum stress).
4.4
Conclusion
By utilizing an atomistic simulation approach based on the first principles reactive
force field ReaxFF, we have investigated the impact of hierarchical structures on the
mechanical response of the most abundant mineral on earth, silica. By incorporating
a hierarchical design concept inspired by diatoms algae, we have modeled two levels
.............................
........
......
(a)
....
...........
-
...
(b)
0#
4w
4
f
4
f
f
410
Ir
'r
'40
f
f
8
f
f
-40
A Ir
(C)
0I-'
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
15 A Foil H Terminated
(d)
1.6
15 A Foil Si surface
1.4
1.2
1.0
(-- 0.8
0.6
0.4
0.2
0.0
0-
0
(e)
8 f
0.2 0.4 0.6 0.8
Engineering strain
0
0.2 0.4 0.6 0.8
Engineering strain
1.50
70
60
i50
c 40
T 30
S20
10
0
1.25
Estrain at max stress
1.00
*sfailure strain [%]
mmax stress [GPa]
[%]
0.75
0.50
0.25
0.00
H Terminated
Si surface
Figure 4-9: Comparison of terminated (a) and non terminated (b) 15 A silica foils. The
stress strain graph is shown for the terminated (c) and non terminated (d). The comparison
of ductility, maximum stress, and strain at maximum stress is shown in (e). Once terminated
the maximum stress decreases by approximately 0.17 GPa and low standard error of 0.5%
compared to 5% of the non termianted case. The standard error of the mean, SEM, is
determined from a total of five simulations for each of the terminated and non terminated
cases. Termination decreases ductility by roughly 1.3%. Each color in the stress strain
graph represent the individual runs.
.....
....
-
.......
............
.......
......
&NA-
(a)
(b)
U'-'-
(C)1.6
0
(U
1.4 _5 A Mesh H Terminated
1.2
1.0
avg. max stress
0.8
0.6
0.4
0.2
g.failure strain
0.0
0 0.5 1 1.5 2 2.5
Engineering strain
t&A&
(d) 1.6
1.4 5 A Mesh Si surface
1.2
1.0
avg. max stress
(U
4a 0.8
(W 0.6
0.4
avg. failure
0.2
strain
0.0
0 0.5 1 1.5 2 2.5
Engineering strain
.50
(e) 250
.25
150
-
100
-
1.00 a
(
-U0.75
4
0
0.50
0.25
50 0
0.00
H Terminated
Si surface
* strain at max stress [%]
nfailure strain [%]
* max stress [GPa]
silica meshes.
Figure 4-10: Comparison of terminated (a) and non terminated (b) 5
The stress strain graph is shown for the terminated (c) and non terminated (d). Once
terminated, the maximum stress increases by approximately 0.15 GPa. The standard error
of the mean, SEM, is determined from a total of five simulations for each of the terminated
and non terminated cases. Termination drastically increases ductility from 122% to 250%,
as shown in (e). Each color in the stress strain graph represent the individual runs.
62
of geometric hierarchies: (1) A nanoscale foil of silica, and (2) a nanoscale silica mesh
composed of interlocking foils. These arrangements were then studied with different
wall widths in order to reveal the effect of size-scaling on their mechanical properties.
Next, we presented a comparison between the mechanical properties of silica meshes
and silica foils. Our findings suggest that higher levels of hierarchy are critical to
greatly improving toughness, and can increase toughness by up to 200% from silica
foil to silica mesh.
We ascribe these magnificent improvements in mechanical properties of the mesh
structures from two competing atomistic mechanisms of deformation; shear and brittle
crack propagation.
In the toughest silica mesh, for example, a crack arrests due
to shearing of another strut, which consequently stiffens the system (see Figure 47).
Interestingly, a size-dependent peaking of toughness was observed for the Si
and SiO 2 meshes; with a maximum toughness of 1.29x109 J/m
= 41 A for the SiO 2 mesh (see, Figure 4-8(b)).
3
corresponding to w
The increased toughness of these
nanostructures also makes them viable candidates for impact-resistant lightweight
structures, and this should be tested in further detail. Another powerful concept
derived from this study is the ability to transform a brittle material into a ductile
one by simply manipulating the geometry of a constituent structure to resemble that
of ordered nanopores, or mesh. Three reasons for the high ductility are three-fold,
(1) a homogeneous distribution of surface stress throughout the entire structure, (2)
a conformational change from rectangular to hexagonal pores, and (3) competing
mechanisms of shear and crack arrest.
In a similar fashion to structural materials found in nature, such as bone, nacre,
or diatom shells, it seems that hierarchy is a cornerstone of nanomaterial design
for superior mechanical properties.
More importantly, the concept of hierarchical
structures made of the same materials (as demonstrated in this article) is fundamental
in order to fully realize the enormous potential of nanomaterial design. To the best
of the authors' knowledge, this paper is the first to elucidate, at the atomistic scale
and with near quantum mechanical accuracy, the complex mechanical response and
failure mechanisms due to implementing hierarchy in silica. The key contribution
of this work is that by introducing structural hierarchies, a weakness can be turned
to strength, that is, an intrinsically strong but brittle material becomes exceedingly
tough, strong, and ductile. The fact that a similar behavior was found in silicon
[34] suggests that this may indeed be a generic design concept that could be used
for many materials The observation that a weakness is turned into strength is also
reminiscent from recent findings of similar behaviors of H-bonds, which are by itself
also highly brittle, weak elements but reach extreme levels of toughness and strength
once arranged in particular hierarchical patterns [47].
Future research lies in addressing the impact of a larger number of hierarchy
levels on the mechanical properties of silica nanostructures, and the effect of water
on hierarchical silica.
Chapter 5
Superductile, wavy silica
nanostructures
The previous chapters focused on the mesh structures found in many diatoms, such
as Bacillariophyceae,and revealed the impact of hierarchy and size on the mechanical
response. The next step in unraveling the interplay between structure and mechanical
response of diatoms lies in understanding their fantastic ability to stretch while still
being intact. The extreme ductility of certain diatom species and communities is a
fascinating attribute, especially since diatoms are mostly made of amorphous silica,
a typically brittle constituent. In this chapter we focus on a particularly interesting
colonial diatom, Ellerbeckia arenariabecause they live in waterfalls and are thus able
to resist significant and continual mechanical stress. These colonies are also able to
elastically stretch up to about 33% [36, 35]! Two possible reasons for this extreme
mechanical response are the intrinsically shaped cell wall and the organic coating,
called mucilage, surrounding the cell wall surface. In this chapter, a new geometry is
analyzed due to its presence in certain diatoms species that are able to elongate and
resist extreme mechanical stress from to the environment. We, therefore, focus on
the corrugated, wavy shape found along the sides of Ellerbeckia arenaria (see Figure
5-1(a)), and propose that this particular shape is essential to providing flexibility
while combining high strength and toughness [32].
5.1
Model geometry
We consider a structural design composed of alpha-quartz crystals. The structure is
a foil or infinitely tall thin wall with varying amplitude and width, resembling a wave,
as seen in Figure 5-1(b). Since the simulation box is periodic, the foil structure can be
thought of an array of waves with a spacing equivalent to the peak-to-peak amplitude.
The z axis has no free surface, and the structure can be described as infinitely tall.
Figure 5-1(c) shows the geometries considered here.
All wave structures have an
equivalent wavelength of 63.5 A. The only parameter varied here is the wall width w
and amplitude A, which range from 20
A to
120 A, and 0
A to
60
A respectively
(see
Figure 5-1(c)). The number of atoms varies from ~ 650 to ~ 7000 for the smallest to
the largest width silica systems. For the wave structures, the largest simulation cell
has dimensions of 177 A x 63.5
5.2
A
x
8.5 A in the x-, y-, and z-direction.
Simulation approach
All structures are equilibrated under the canonical ensemble at 300 K for a time of
10 ps and then loaded under uniaxial strain loading along the [1 2 0] direction as
shown in Figure 1(b), at a strain rate of Ix1010 s1 at 300 K. The system has periodic
boundary conditions in all three directions and the temperature is controlled by the
Berendsen thermostat [7]. Deformation is applied by uniaxially increasing the size
of the periodic simulation cell in the loading direction only, while keeping all other
dimensions of the simulation cell constant. We use a time step of 0.2 femto-seconds.
The initial, unstrained silica structure is shown in Figure 5-1(c).
Aside from the
variations in the geometry, all simulations are carried out under identical conditions,
enabling us to perform a systematic comparison.
5.3
Results and discussion
Here we present our analysis on the effect of altering the amplitude and wall width on
the mechanical properties of the silica wave structures (see Figure 5-1(c) for the two
..............
(a)
(b)
Sit2uWave
structure
y [120]
100]
z [001]
(c)
Width
Amplitude
60 A
30
20 A
51 A
I'll
80 A
120 A
A
15 A
0A
Figure 5-1: Panel (a): A colonial diatom, Ellerbeckia arenaria,lives in waterfalls and
contains girdle bands with intricate patterns. Specifically, a wave shape can be seen, which
might be an important contribution to the elastic response of approximately 33%, as observed through AFM experiments [36]. Panel (b): Initial geometry of a bioinspired silica
structure used in our simulations, illustrating the wall width and amplitude (w and A,
definitions indicated in the structure). Panel (c): Initial geometry of all wave structures
considered here, illustrating the range of variation in amplitude and width. Images reprinted
from Ref. [36], Gebeshuber, et al. Journal of Nanoscience and Nanotechnology 2005 [permission pending].
geometries considered). As shown in Figure 5-1(c), the wall width w and amplitude
A, which range from 20 A to 120 A, and 0 A to 60
A respectively.
These structures
resemble those found in some diatoms, such as Ellerbeckia arenaria.
The stress strain response for all structures is shown in Figure 5-2. Here we observe
a maximum strain of 270% for the structures with largest amplitude, 60 A and w of
51 A and 80 A. Interestingly, the initial modulus of these structures is roughly ten
times lower than those with lower amplitude. For structures with largest width and
lowest amplitude, the greatest modulus and maximum stress are reached: E = 14.4
GPa and om,= 5 GPa. The general trend is for decreasing failure strain, and greater
modulus and maximum stress as the wall width is increased and the amplitude is
lowered.
Next, we analyze the effect of altering the wall width and amplitude on the toughness, ductility, maximum stress, and modulus, as seen in Figure 5-3. The toughest
response is seen for amplitudes below 30
A and
greater than 51 A. It is important to
note that these structures are defect free, and thus the upper limit on toughness can
diverge from experimental ones. Ductility is highest, reaching ~270%, for amplitudes
of 60
A, and 51 A
A of and 80 A
w
w
80 A. The largest maximum stress is found for 0
120
A. Toughness versus
A
A
15
modulus is compared in Figure 5-3 (c),
showing that they are both positively correlated for decreasing amplitude. However,
the increased toughness and modulus comes at the expense of ductility. The structure
with greatest toughness of 1.3 GJ/m 3 corresponds with a relatively low ductility of
~50%.
The observed deformation mechanisms are closely linked to the mechanical response of each structure. In Figure 5-4 we compare the Von Mises stress fields of
three structures and the corresponding deformation mechanisms during failure which
correlate with three distinct mechanical responses: high toughness, high ductility,
and high stress. The structure with highest toughness is shown in Figure 5-4 (a), and
contains a significant portion of the stress to form a straight line, and lower stress near
the curved regions. As failure occurs, cracking initiates near the corners and propagates in a diagonal shearing fashion. Void formation dictates the path of cracking,
.........
....
...........
(a) 5.0
4.5
4.0
S3.5
35
0-q
03.0
2.5
2.0
1.5
1.0
0.5
0.0
(b) 5.0
4.5
4.0
-A=OA, w=20A
A=15A, w=20A
- A=30A, w=20A
- A=60A, w=20A
3.5
W
3.0
-2.5
S2.0
S -A=30A,
A=0A, w=51A
-A=15A, w=51A
-1
w=51A
-A=60A, w=-51 A
1.5
111
0 0.5 1 1.5 2 2.5 3
Engineering strain
"
IIIIIIIIII
1.0
0.5
0.0
0 0.5 1 1.5 2 2.5 3
Engineering strain
(c) 5.0
4.5
4.0
(d) 5.0
4.5
4.0
0-3.5
S3.5
~3.0
3.0
2.5
-2.5
2.0
2.0
1.5
1.5
1.0
0.5
1.0
0.5
0.0
0.0
0 0.5 1 1.5 2 2.5 3
Engineering strain
0 0.5 1 1.5 2 2.5
Engineering strain
Figure 5-2: Stress-strain graph of silica wave structures, for w = 20 A panel (a), w = 51 A
panel (b), w = 80 A panel (c), and w = 120 A panel (d). Panel (a): The structures of lowest
width, 20 A, resist a maximum stress of 1.5 GPa, seen in A = 20 A. For A = 60 A, w = 20 A,
the structure fails immediately upon loading. In A = 30 A, w = 20 A, the structure unfolds
and straightens, thus allowing for steep increase and plateau in stress at 100% strain. Panel
(b): The structures with amplitudes of 15 A and 30 A increase in brittleness when the wall
width is increased. However, for A = 30 A the ductility is drastically increased to 283%
strain. The gradual increase in stress indicates subtle unfolding and eventual straightening
of the structure. Panel (c): As wall widths are increased to 80 A the maximum stress rises
for all structures, but also fails sooner than structures with smaller widths. Panel (d): The
highest stress is observed, reaching 5 GPa for A = 15 A, w = 120 A. However, brittleness
increases dramatically and the unfolding mechanisms is no longer observed for A = 60 A, w
= 120 A, which fails at 130% strain. Failure mechanisms characteristic of the most brittle
structures, such as A = 15 A, w = 120 A, are crack and shear, along with void formations
that coalesce with other voids until the structure is no longer intact. Bulk silica has many
defects which allow for its brittle nature. We performed a simulation of periodic bulk silica
with a penny shaped crack 5 A wide and 120 A long, that extended through 120 A of silica,
that failed at 12% strain and reached a maximum stress of 5.3 GPa.
69
..........
::::::::::::::::::::..
1.11
(a)
-
-
-
11
-
60
Toughness
[GJ/m 3]
30
4-a
r
()60
0.5-1
E 0-0.5
E
0t
15
0
60
-a.- 154
0 o
CN
P
-
U)
0
0o
Width [A]
Width [A]
(d)1.4
Maximum
Stress [GPa]
0 4-5
* 3-4
30
<E
0
CO
CO
Width [A]
(c)
n200-300
100-200
E 0-100
E
0
C'J
Ductility [%]
30
N1-1.5
154
-
.
!
2-3
* 1-2
. O-1
E 1.2
=5 1.0
0 0.8
C
S0.6
.E 0.4
0.2
co
0
A
* w=20A
A
w=51A
_w=80A
* w=120A
-"
**
0.0
T I
I
I
I
I
I
I
10
5
Maximum Stress [GPa]
0
Figure 5-3: Panel (a): Toughness map, showing that greatest toughness is achieved for
structure with A = 15 A, w = 120 A. Panel (b): Ductility map. Structures with greatest
ductility have the highest amplitude, reaching up to 283%. However, as the amplitude
becomes 0-15 A, ductility is generally below 100%. Panel (c): Maximum stress map, showing
that structures with lowest amplitude and large width, such as A = 15 A, w = 120 A, reach
a stress up to 5 GPa. Panel (d): Nonlinear relationship between modulus and maximum
stress. The structure with greatest toughness is A = 15 A, w = 120 A.
while the shearing effect breaks off the outermost surface of the voids in a sequential manner, similar to a beading mechanism. For structures with highest ductility,
the deformation path is unfolding and finally beading. The unfolding mechanism
encompasses the straightening of the central, or core structure, while the wave peaks
remain in the initial conformation and open up slightly. Rotation of the core region is
enhanced by single void formation near the corners. The core region is defined as the
area bounded by the inner peaks of the wave. For structures with maximum stress,
the deformation mechanism is mainly cracking, with very little shear deformation.
(a) A=15A w=120A
E
= 36%
A=A w=120A
A=60A w=51A
E = 36%
E = 242%
Mises stress
(b) A=15A w=120A
E =
56%
A=OA w=120A
E = 45%
A=60A w=51A
E=
283%
20
16
121
8
4
0
Figure 5-4:
Von Mises stress field for structures at maximum stress, panel (a), and at
failure, panel (b). Panel (a): The structure with A = 15 A, w = 120 A contains a significant
portion of the stress to form a straight line, and lower stress near the curved regions. For A
= 0 A, w = 120 Athe stress is relatively homogeneous, with highest stress on the surface.
For A = 60 A, w = 51 A high stress is concentrated along the extended, vertical ligaments.
Panel (b): High stress is observed near the failure process zone for the structure with A
= 15 A, w = 120 A. The structure forms voids near the corner and shearing occurs next.
For A = 0 A, w = 120 A, failure is mainly crack formation, while unfolding and beading is
observed for A = 0 A, w = 120 A.
......
.....
.......
.....
......
I...
.................
....
..
......
.....
Next, we map the regions of best performance - ductility, toughness, and maximum stress - and its relationship with the amplitude and width, as seen in Figure
5-5. For structures with high amplitude and low width, greatest ductility is achieved;
whereas low amplitude and high width is yields the greatest strength. The structures
of highest toughness have a width and amplitude bounded by the regions of highest
ductility and stress. Interestingly, it is a balance of both geometric parameters (A and
w) and a combination of deformation mechanisms (unfolding, shearing, and cracking)
which allow for greatest toughness. Furthermore, this concept of geometric effects on
the stress-strain behavior are displayed in Figure 5-6.
60
(best ductility)
30
E15
cracking
(best str
0
10
Figure 5-5:
20
51
80
Width [A]
120
Performance map, showing the regions where optimum toughness, ductility,
and strength are located with respect to width and amplitude.
An analogy to protein structures can be drawn, wherein sacrificial bonds and
..
....................
I
Thin & wavy
Wide & wavy
Wide & straight
Wide & straight
Co
C')
Wide & wavy
ci)
Thin & wavy
Strain
Figure 5-6: Schematic of stress strain response for tensile deformation of different morphologies of silica waves. Thin and wavy structures provide greatest ductility, while wide
and straight structure provide high stress at the cost of ductility. However, when combining
wavy and wide morphologies, significant toughness is gained.
hidden lengths are responsible for enhanced toughness [27, 81, 16]. These sacrificial
bonds are weaker than the carbon backbone, but stronger than van der Waals or
hydrogen bonds, and allow for saw-tooth shape force-extension curves. Sulfate bonds
are a great example of sacrificial bonds in systems containing DOPA, a common amino
acid found in biological adhesives. As each sacrificial bond breaks, energy is released
in the form of heat and a regional unfolding, or uncoiling of a hidden length segment,
occurs. This process is repeated until all the sacrificial bonds are broken and the
structure is completely unfolded. Only then will the carbon backbone break, resulting
in the highest peak of stress. The interplay of multiple failure zones and deformation
mechanisms is strikingly similar to those found in the silica wave structures. A link
is made between the sacrificial bonds found in proteins, and the shearing mechanism
found in the silica wave. The catastrophic carbon backbone failure is also analogous to
cracking in a silica system. When integrating these multiple mechanisms, a universal
concept of enhancing toughness is achieved.
5.4
Conclusion
In summary, we have investigated the fundamental impact of wave structure geometries on the mechanical response of silica, by utilizing an atomistic simulation
approach based on the first principles reactive force field ReaxFF. Specifically, we are
able to demonstrate that the ductility of silica can reach to approximately 270%, by
increasing the amplitude to 60 A and maintaining the width at 51 A. This is achieved
by unfolding mechanisms and straightening of the structure, similar to the uncoiling
of hidden length from a convoluted protein. The structures with greatest toughness,
reaching values of up to 1.3 GJ/m 3 , have a 15
A amplitude
and a 120 A width. And
finally, greatest strength is obtained from straight and widest, w = 120 A, structures.
Indeed, the study in this chapter is an important step towards revealing the broad
range of mechanical properties achieved through altering specific geometric shapes
(s.a. mesh to wavy), and will undoubtedly pave a conceptual figure of merit for future
nanoscale structural designs. Discovering that silica can indeed become extremely
ductile opens opportunities for industrial applications, such as flexible nanoscale processors, and offers new incentives, in terms of mechanical response, for the incorporation and fabrication of wavy silica structures.
76
Chapter 6
Conclusion
The ability to improve upon multiple mechanical properties, such as toughness,
strength, and ductility, is extremely important when designing future nanoscale materials. Altering the mechanical properties of one of the most brittle and abundant
minerals on earth, silica and silicon, allows a new window of opportunity for humanity to create applications and reinvent materials once thought to be impossible.
The transferability of the concept allowing for massive transformation of mechanical
responses, such as brittle to ductile or weak to tough, through geometric alterations
at the nanoscale, is another profound discovery that will undoubtedly unleash a new
paradigm in the way materials are designed and applied. Indeed, the culmination
of materials design is to maintain environmental sustainability, infrastructure superiority, multifunctional capacity, and economic feasibility. Nanoscale materials implemented through design and fabrication concepts found in biology, such as in diatom
algae, bone, and sea sponges, hold the promise of providing these advantages.
6.1
Summary of findings
Revealing the intricate interplay between structure and mechanical response of diatoms requires delicate and systematic analysis. The previous chapters established a
framework on size effects, hierarchy, and shape, on the mechanical response of silicon
and silica structures. The main findings are summarized as follows:
"
Silicon and silica meshes yield highly tunable mechanical properties through
alteration of their width, as shown in Figure 4-8 . A region of optimum toughness is observed to lie between w=20
A
and w=50 A. Crack arrest and shear
mechanisms allow for the high toughness, as shown in Figure 4-7.
" Hierarchical structures, such as meshes, have superior toughness and ductility
when compared to their constituent counterparts, foils, as shown in Figure 42. The foils lack the multiple mechanisms of failure that are observed in the
meshes.
" Wavy silica structures are able to reach extremely high ductilities of up to
~ 270%, as shown in Figures 5-3 and 5-4, through unfolding mechanisms and
straightening of the structure, similar to the uncoiling of hidden length from a
convoluted protein.
6.2
Discussion and future research
Several challenges remain in the form of fabricating and more accurately modeling
these structures. For example, biomineralization from self assembling proteins which
guide silica precipitation has been studied [53, 52, 86, 85]. However, the ability to
synthesize complex and hierarchical structures still remains challenging. The recent
determination of certain diatom genetic sequences will further the understanding of
accurately controlling and fabricating silica structures. In terms of modeling these
systems, a key consideration is the effect of surface reconstruction on the mechanical
properties. However, surface reconstruction occurs on timescales that are intractable
for many atomistic methodologies, such as ReaxFF or quantum based approaches.
Future research could be geared toward atomistic simulations on the deformation
and failure of different morphologies found in diatom species. Moreover, mineralized
structures are found in many other biological systems, such as deep see sponges [2],
which could be studied using a similar molecular approach.
Another important step is reaching a greater convergence between actual diatom
frustules and those modeled. Key challenges are reaching greater size scales, incorporating organic material, amorphization, and surface termination. The size scale issue
can be generally overcome with coarse graining, or utilizing massive supercomputers
on the order of hundreds of cpu's with ReaxFF. With larger systems, more complex
shapes can be modeled, such as incorporating different shapes throughout the z axis.
Perhaps a more complicated challenge is the addition of organic material, such as proteins, within the silica structure. The existing ReaxFF forcefield that models both
organics and silica is limited to glyoxal, and does not encompass nitrogen bonds, an
element found in many organic structures, such as collagen [51]. Once the adequate
forcefield is developed, proteins such as silaffins and collagen could be added to the
silica. As previous studies have mentioned, proteins within diatoms are found in their
adhesives and enable self assembling [22]. Amorphization of silica is another critical
concept that should be explored, since it could affect the mechanical properties and
is also found in diatoms. Another avenue for further research is surface termination,
as it occurs when silica is exposed to water, and does affect the mechanical response
of silica structures, as observed in the preliminary investigation in this thesis. Future
simulations of surface terminated structures will encompass larger systems in order
to more fully capture the effect at different length scales.
80
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